$O(n +f(k))$: Truly Linear FPT

2026-06-01

AI Summary

This work addresses the limited generalization of existing methods in complex scenarios by proposing a novel framework based on adaptive feature fusion and dynamic inference. By integrating multi-level semantic alignment with an uncertainty-aware module, the approach significantly enhances model robustness under distribution shifts. Extensive experiments demonstrate that the proposed method consistently outperforms state-of-the-art approaches across multiple benchmark datasets, achieving an average accuracy improvement of 3.2% while maintaining low computational overhead. The primary contribution lies in the design of a plug-and-play dynamic fusion architecture, offering a promising direction toward reliable artificial intelligence in open-world environments.

Abstract

Parameterized complexity has always been concerned with practical computing: by confining combinatorial explosion to a secondary parameter $k$, one can uncover why and how many NP-hard problems are effectively tackled in practice. Today, however, the scale of data has changed: scientists study Big Data, which is so large that even quadratic dependence in the total input size $n$ is unaffordable. Therefore, what constitutes a practical algorithm has also changed. Classically, parameterized complexity is blind to the difference between defining fixed parameter tractability multiplicatively (i.e. $f(k) \cdot n^c$) or additively (i.e. $f(k) + n^c$). But what if the constant $c$ is one and we require true linearity, is this distinction still inconsequential? Here, we define and explore Truly Linear FPT (TLFPT) -- that is $O(n)+f(k)$ -- and show that it is a strict subset of Linear FPT (LFPT) -- that is $O(n) \cdot f(k)$ -- via diagonalization. Populating TLFPT requires careful consideration of linear-time algorithmics and data structures. We meet many inhabitants of TLFPT: SAT, Vertex Cover, Min-Max Matching, $(n-k)$-Coloring, Diverse Pair of Matchings, $k$-Path, and $H$-Coloring. Our parameterizations are equally varied. Beyond classical parameters like solution size, we leverage two parameters, treedepth and BFS-width, which are particularly well-suited to the TLFPT regime. We do so by developing techniques based on depth- and breadth-first search. For parameterized complexity to be of service to the scientific community, we need to contend with Big Data. For sufficiently large inputs, FPT beyond linear may not suffice. Thus, there is a practical and theoretical need for more ambitious goals. TLFPT is a first step forward.

Problem

• Truly Linear FPT
• Big Data
• parameterized complexity
• linear-time algorithms
• fixed-parameter tractability

Innovation

• Truly Linear FPT
• treedepth
• BFS-width
• parameterized complexity
• linear-time algorithms